#### Abstract

In this paper, we study the notion of approximate biprojectivity and left -biprojectivity of some Banach algebras, where is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra implies that is compact. Moreover, we investigate left -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra is left -biprojective if and only if is compact, where is a hypergroup. We also study the approximate biflatness and left -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

#### 1. Introduction and Preliminaries

Hypergroups are a suitable generalization of classical locally compact groups. In classical setting, the convolution of two point mass measures is a point mass measure, while in hypergroup structure, the convolution of two point mass measures is a probability measure with compact support. The study of hypergroups was initiated in the 1970s by Dunkl [1], Jewwet [2], and Spector [3], each of them in various axioms. However, in this paper, we will base our work on Jewett’s axioms in [2].

Biprojectivity is an important homological notion that arises naturally in Helemskii’s works in the 1980s; interested readers can refer to his comprehensive book [4]. Biprojectivity of some well-known Banach algebras associated to locally compact groups, such as group algebras and measure algebras, is studied in [4, 5]. Biprojectivity of the hypergroup algebra is studied in [6]. As a generalization of this notion, Y. Zhang in [7] introduced the notion of approximate biprojectivity. Indeed, a Banach algebra is called approximately biprojective if there exists a net of continuous -bimodule morphism from into such that for every , where is the diagonal operator defined by . For recent works about this concept, refer to [8].

Throughout the paper, stands for the set of all nonzero multiplicative linear functionals on . Kaniuth et al. [9] introduced the notion of left -amenable Banach algebras as a generalization of the notion of amenable Banach algebras introduced by Johnson in [10]. A Banach algebra is called left -amenable if every derivation from into is inner, for every Banach -bimodule with the left module action for all and .

Hu et al. in [11] defined the notion of left -contractibility for Banach algebras. Following [12], a Banach algebra is called left -contractible, where , if there exists such that and , for every . For a locally compact group , it is shown that left -contractibility of (or ) is equivalent to compactness of (Theorem 6.1 in [15]).

Motivated by these considerations, the first author defined the homological notion of left -biprojectivity for Banach algebras (see, e.g., [13]). Here is the definition of his new notion. A Banach algebra is called left -biprojective, where , if there exists a bounded linear map such that

The right case can be similarly defined.

The first and the second authors in [13] explained the relation between left -contractibility and left -biprojectivity. They proved that left -contractibility implies left -biprojectivity, and the converse is valid if is commutative or has a left approximate identity.

We give some brief backgrounds on hypergroups and their related algebras and establish our notation; for details, see [14]. Let be a locally compact Hausdorff space and denote the set of all bounded complex Radon measures on , where the norm of each measure is the total variation . Also, we denote by the set of all nonempty compact subsets of and equipped this space with the Michael topology, that is, the topology generated by the setsfor open subsets and in .

The space is called a hypergroup if there is a convolution , an involution on , and an element (called the identity element) such that the following holds:(i) is a Banach algebra.(ii) is a probability measure with compact support.(iii)The map is continuous from into equipped with the weak topology.(iv)The map is continuous from into .(v)For each , .(vi)The mapping is a homeomorphism on of period 2 and and if and only if for each . Here, the measure is given by for all Borel subsets .

A hypergroup is called commutative if for all . In [2, 3], it is proved that every commutative or compact hypergroup has a unique (left) Haar measure. The existence and the uniqueness of left Haar measure on a general locally compact hypergroup were proved recently by Chapovsky in [15]. Throughout this paper, we assume that is a hypergroup with a unique Haar measure . That is,for every Borel measurable function on . Then, with the involution and the convolutionis a Banach -algebra, where for every . A nonzero bounded continuous function is called a character on if for every . The set of all characters on will be denoted by , i.e.,

For , define on via

It may be observed that , and note that there is at least one character on , namely, the augmentation character . Further, if is commutative, it is well-known that there is no other character on ; that is, . For more details, see Section 2.2 in [3].

In the present paper, we show that approximate biprojectivity of hypergroup algebra implies that is compact. After that, we study left -biprojectivity of general abstract Segal algebras with respect to the hypergroup algebra . As a result (with a mild condition), we conclude that the abstract Segal algebra is left -biprojective if and only if is compact. We also study approximate biflatness and left -biflatness of some hypergroup algebras in terms of amenability of their related hypergroups.

#### 2. Approximate Biprojectivity and Left -Biprojectivity of Hypergroup Algebras

Recall that if is a Banach algebra and is a closed two-sided ideal in , then for each such that , the map is a character on .

Proposition 1. *Let be a Banach algebra, , and let be a closed two-sided ideal of such that . If is left -biprojective, then is left -biprojective .*

Proof. Since is left -biprojective, there is a bounded linear map such that and for every . Consider the quotient map . It may be noted that

By the assumption , we have . Thus, can be dropped on . Define by . It may be observed that

Moreover,and also

Hence, is left -biprojective.

Corollary 1. *Let be a Banach algebra with a left approximate identity and . If is a closed two-sided ideal of and is left -biprojective, then is left -biprojective.*

In the group case, it is well-known that the group algebra is biprojective if and only if is compact (see, for example, [4]). In the similar way, in Theorem 3.1 in [6], it is shown that if the hypergroup algebra is biprojective, then is compact. In the following, we give a generalization of this result and characterize approximate biprojectivity of the hypergroup algebra .

Theorem 1. *Let be a locally compact hypergroup. If the hypergroup algebra is approximately biprojective, then is compact.*

Proof. Let be approximately biprojective. Since has a bounded approximate identity (Theorem 1.6.15 in [3]), the hypergroup algebra is left -contractible for every (Theorem 3.9 in [20]). Consider the augmentation character on via

So, there exists satisfying and for every . Choose such that . It is worth noting that

Since , we have . Thus, . It follows that is a constant function. Therefore, . The latter implies that is compact (page 40 in [3]).

We study left -biprojectivity of abstract Segal algebras. So, we start with definition of abstract Segal algebras.

*Definition 1. *Let be a Banach algebra with norm . We say that a Banach algebra with norm is an abstract Segal algebra with respect to if(i) is a dense left ideal in A(ii)There exists such that for every (iii)There exists such that for every and Moreover, we say that is a *symmetric abstract Segal algebras* if is a two-sided ideal of and there exists such that for every and .

For more details about abstract Segal algebras, see [16]. We note that there are some abstract Segal algebras which are not symmetric. Homological and cohomological properties of abstract Segal algebras have been studied in many papers (see, for example, [17–19]). Recall that it is known that (see Lemma 2.2 in [1]). It is worthwhile to mention that Essmaili et al. in [6] studied right -biprojectivity (they called it condition ) of Banach algebras associated with a hypergroup, especially symmetric Segal algebras. As a main result, they showed that if is a commutative hypergroup and is a symmetric abstract Segal algebra with respect to , then is right -biprojective if and only if is compact, where . In the following, using Theorem 1, we extend the left version of (Corollary 3.11 in [6]).

Theorem 2. *Let be a locally compact hypergroup, , and let be an abstract Segal algebra with respect to which possess a left approximate identity. Then, the following statements are equivalent:*(i)* is left -biprojective*(ii)* is compact*

Proof. Suppose that is left -biprojective. Applying Proposition 1 in [18], is left -contractible. By Proposition 2.5 in [1], is left -contractible. By similar argument as in the proof of Theorem 1, is compact.

Conversely, let be compact with the normalized Haar measure (page 40 in [3]). Then, for each , we havefor some . So, . Put . We claim that and for every . To see this,and also

So, is left -contractible. Using Proposition 2.5 in [1] again, is left -contractible. Hence, is left -biprojective (Lemma 1 in [18]).

#### 3. Approximate Biflatness and Left -Biflatness of Hypergroup Algebras

In this section, we study approximate biflatness and left -biflatness of some algebras related to a hypergroup.

We remind that a Banach algebra is called approximately biflat if there exists a net of -bimodule morphisms from into such that . Here, stands for the weak operator topology (see [20]).

Following [21], a locally compact hypergroup is called left amenable if there exists a left invariant mean on . That is, a bounded linear functional such that and for all and .

Proposition 2. *Let be a locally compact hypergroup and let be an abstract Segal algebra with respect to which possess a left approximate identity. If is approximately biflat, then is left amenable.*

Proof. Suppose that is approximately biflat and fix a character . Since has a left approximate identity, by Proposition 2. 4 in [22], is left -amenable. Now, Proposition 2.3 in [1] follows that is left -amenable. Applying Theorem 3.5 in [14], we conclude that is left amenable.

Corollary 2. *Let be a locally compact hypergroup and let be an abstract Segal algebra with respect to which possess a left approximate identity. If is approximately biflat, then is left amenable.*

Proof. Fix a character . Since has a left approximate identity, similar to the arguments as in the proof of (Theorem 2.2 in [23]), approximately biflatness of follows that is left -amenable. It deduces that is left -amenable. Thus, by Theorem 3.5 in [14], is left amenable.

Let be a Banach algebra and . Then, is called left -biflat if there exists a bounded linear map such that

Here, is a unique extension of which is defined by for all . The right case can be defined similarly. For further information, see [24].

Theorem 3. *Let be a locally compact hypergroup, , and let be an abstract Segal algebra with respect to which possess a left approximate identity. The following statements are equivalent:*(i)* is left -biflat*(ii)* is left amenable*(iii)* is left -biflat*

Proof. (i) (ii) Let be left -biflat. Since has a left approximate identity, by Lemma 2.1 in [21], is left -amenable. It follows that is left -amenable. Now, applying Theorem 3.5 in [14], we conclude that is left amenable.

(ii) (iii) Let be a left amenable hypergroup. Then, by Theorem 3.5 in [14], is left -amenable. Now, by Proposition 2.3 in [1], is left -amenable. Using Proposition 3.4 in [13], is left -amenable and then Lemma 2.3 in [25] implies that is left -biflat.

(iii) (i) Suppose that is left -biflat. Then, by Theorem 2.2 in [21], is left -biflat.

Suppose that and is a locally compact hypergroup. Set with the norm

becomes an abstract Segal algebra with respect to . It is known that possess a left approximate identity (see [2]).

The following corollary is an easy consequence of Proposition 2 and Corollary 2.

Corollary 3. *Let be a locally compact hypergroup and . If (or ) is approximately biflat, then is left amenable.*

*Example 1. *In this example, we give a Banach algebra which is left -biflat but it is not right -biflat.

Let be a Banach space with and be a nonzero functional such that . Define a multiplication in via , for all . It is easy to see that is a Banach algebra and . We denote the unitization of with . It is known that is a closed ideal in . Moreover, has an extension to , that is, which is given by for all and . We claim that is left -biflat but it is not right -biflat. To see this, we know that is left -amenable, applying Lemma 3.2 in [13], it follows that is left -amenable. It gives that is left -biflat. Now, assume conversely that is right -biflat. Since is unital, has an element such that and , for all . Following the similar arguments as in Theorem 1.4 in [13], we have a bounded net in such that and , for all . Pick an element in such that . Replace with , we may suppose that is a bounded net in such that and , for all . However,for all . Let be any element in and put it at above fact. It follows that . So, is an isomorphism. Therefore, which is a contradiction.

#### Data Availability

The data that support the findings of this study are available from all the authors.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The first author is thankful to Ilam University, for their support.